Heredity 59 (1987) 19—28 The Genetical Society of Great Britain Received 5 August 1986
Biochemical heterozygosity and phenotypic variability of polygenic traits
Ranajit Chakraborty Center for Demographic and Population Genetics, University of Texas Graduate School of Biomedical Sciences, P.O. Box 20334, Houston, Texas 77225, U.S.A.
Using the theory of additive genetic variability of a polygenic trait, it is shown than an individual's heterozygosity at the loci governing the trait cannot be determined accurately from observations on phenotypes alone. Furthermore, the negative association between heterozygosity and phenotypic variance, and a positive correlation of the frequency of the modal class of a phenotypic trait and the extent of heterozygosity can be explained by additive allelic effects. It is argued that while the number of heterozygous loci in an individual may not be a good indicator of the individual's genomic heterozygosity, there is evidence that some of the biochemical loci may reflect genetic variation at the loci controlling phenotypic polymorphism. Thus the observed relationship between biochemical heterozygosity and phenotypic variance may not constitute hard evidence of heterosis, overdominance, or associative overdominance.
INTRODUCTION invoked to explain the observation of positive associations between the heterozygosity and the Allozymicvariations in natural populations, lack of phenotypic variability (see e.g., the referen- studied over the past two decades, have now con- ces cited in the above two reviews, and Livshits vincingly established that genetic variation is gen- and Kobyliansky, 1985). erally ubiquitous in almost all taxa. Recent The main features of the positive results of such molecular data showing genic variations at the association studies may be summarised into three DNA level have strengthened this view (Nei, 1987). observations. (a) When heterozygosity is measured Yet, the controversy regarding the mechanistic by the number of loci at which an individual is explanations for the production and maintainance heterozygous, increased heterozygosity is often of genetic polymorphism is still unresolved, since associated with a decreased phenotypic variability. direct searches for selection at any individual locus (b) The frequency of modal phenotypes is posi- have produced ambiguous results at best. More tively correlated with the heterozygosity levels of recently, therefore, attempts have been made to individuals. (c) Parental fitness is positively associ- relate the heterozygosity levels detected by multi- ated with the expected degree of heterozygosity ple loci with variabilities in morphological or among offspring. None of these findings are, fitness related traits in order to test heterotic effects however, universal, since exceptions are found in of the loci studied. Critical appraisals of the use repeated studies in the same population (Pierce of allozyme data in this regard have been made in and Mitton, 1982), and in studies involving popu- recent reviews of Mitton and Grant (1984) and lations of different evolutionary origin (e.g., Zouros and Foltz (1987). These reviews and their Gottlieb, 1977; Handford, 1980; Knowles and cited references indicate that the efforts to relate Mitton, 1980; Knowles and Grant, 1981; Mitton allozymic heterozygosity with phenotypic variabil- et al., 1981; Chakraborty et a!., 1986). ity do not always produce positive results. Yet, in In spite of these discordant results, the above many organisms, from plants to human, associ- features of association of enzymic heterozygosity ations between these two parameters suggest that with phenotypic variability demand satisfactory heterotic effects may be detectable by studying mechanistic explanation. The interpretation of interactions of multiple loci together. The concept selective differentials among individuals involve of developmental homeostasis has also been the assumption that the morphological traits used 20 R. CHAKRABORTY in determining phenotypic variability are geneti- A) and q, (for the allele B), where q =1—p1. The cally controlled, and the biochemical loci heterozygosity for each individual is measured by employed reflect genetic variation at the loci con- the number of heterozygous loci, if there are trolling such traits. It is true that genetic factors genotypic scoring techniques available for these are involved in morphological variation, and loci. Let Y denote the number of heterozygous fitness may be genetically controlled. Yet, the role loci in a random individual, and X denote the of non-genetic modifiers in determining genotypic value of the individual. We can write X phenotypes of such traits cannot be totally ignored. and Y as Furthermore, the task is more complicated, since the number and nature of genetic loci involved in x=xI+x2+. . .+xn (1 a) morphological or fitness related traits are not and known, and their relationships with structural bio- chemical loci are not clear. Y=Y1+Y2+...+Yn, (ib) The purpose of this paper is to show that the where the bivariate distribution of X, and 1', is classification of individuals into different heterozy- given by gosity classes by phenotypes of a polygenic trait is quite error prone, even if the trait is under complete genetic control. It is also shown that the X1 Y, Genotype Probability increased frequency of modal phenotypes in highly heterozygous individuals (and the consequent 2 0 AA p decrease of phenotypic variability in them) is a 1 1 A,B1 2pq1 direct consequence of additivity of allelic effects 0 0 B.B1 q of a polygenic trait. Lastly, the question of inter- dependence of heterozygosity at biochemical and Under the assumption that the loci are indepen- phenotypic level is addressed by reviewing the dently segregating in the population, the joint dis- theory of predictability of genomic heterozygosity tribution of (X, Y) can be evaluated by the bivari- using electrophoretic markers, in view of the recent ate probability generating function method (Feller, comments of Zouros and Foltz (1987) and Smouse 1950); i.e., the probability that X =rand Y= k is (1986). given by Prob(X=r, Y=k) DISTRIBUTIONOF HETEROZYGOSITY BY = Coefficientof ss in G(s1, s2), PHENOTYPE CLASSES where Inrelation to the studies of morphological vari- ation and the heterozygous nature of individuals, G(s1, s2) =H(q+2pq1s1s2+ps), (2) it is common to identify the heterozygous status of individuals by their phenotypic scores for a for r =0,1,2, .. . , 2n;k 0, 1,2,..., n, and s1, s2 heritable polygenic trait (see e.g., Livshits and are two arbitrary variables taking values between Kobyliansky, 1984; 1985; Kobyliansky and —1 and 1. Livshits, 1985). This is based on the supposition Note that the marginal distributions of X and that the phenotypic value of a morphological trait Y can be obtained by evaluating the coefficients is determined by polygenic effects of the loci of s in G(s1, 1) and s in G(1, s2), respectively. controlling the trait, and the allelic effects are Chakraborty (1981) provided a computational additive. Nevertheless, there has been no attempt algorithm for computing the distribution of Y, and to check this analytically. To address this question, the same technique also applies to X. Since for the let us consider a quantitative trait controlled by n quantitative phenotypes, the distribution of Y can- loci, at each of which there are two segregating not be generally observed directly, one might be alleles (say, A, and B, for the ith locus; i = interested in drawing inference regarding it by the 1,2,...,n). For simplicity, let us assume that the conditional distribution of Y (number of heterozy- allelic effects are all additive. Without loss of gen- gous loci) given an observation on X (the erality we can assume that the allelic affects of A phenotypic score). In practice, however, this is is one and the effect of B, is zero, for all i complicated by the effects of non-genetic environ- 1, 2,..., n. We shall further assume that the allele mental factors which may modify the genotypic frequencies at the ith locus are p (for the allele score X. For simplicity, if we assume that X is HETEROZYGOSITY AND PHENOTYPIC VARIANCE 21 directly observable and is unaffected by environ- Itis interesting to note that when all loci con- mental factors, the conditional distribution of Y trolling the quantitative trait X have the same allele given X =ris given by frequencies, the conditional distribution of Y given X does not depend on the allele frequency Prob(Y=kIX=r) p. Furthermore, since equation (6) is defined only Coefficientof ss in G(s1, s2) for even values of r —k,for a fixed value of r(0 — r 2n), k can take values from 0 to mm. (r, 2n —r), Coefficientof s in G(s, 1) such that r —kis always even. Equations (3) and for 0 r2n, and 0 k n. (6) are instructive enough to show that the For the case where the gene frequencies at all classification of individuals into heterozygosity loci are equal, i.e., p, =p for all I, it is easy to show classes by phenotypic scores can be quite mis- that leading. This is numerically demonstrated in table 1, where the conditional distribution of Y (the Prob (X =r)= (2n) prq2n_r (4a) number of heterozygous loci) for different genotypic values (X) are shown for the case where the allele frequencies are identical for all loci (the and exact value is irrelevant, since eqn. (6) is indepen- dent ofp). Prob(Y= k)=()(2pq)k(p2+q2). Two situations are illustrated with table 1, the (4b) number of loci (n) being 5 and 10. In both cases it is seen that the central phenotype (X = 5 and Furthermore,in this case the multi-locus 10) can have heterozygosity at less than half of the genotype of any individual can be represented by loci with probability approaching 1/4 or more. It atriplet(n1, n2, n3),where n1, n2,and n3represent can be shown from eqn. (3) or (6), that the mean the numbers of loci exhibiting genotypes AA, AB, number of heterozygous loci increases gradually and BB, respectively; with n1 + n2 + n3 = n. Thus, from extreme to central phenotypes, numerical the conditional probability of equation (3) illustrations of which are given in fig. 1. However, becomes since the distributions of the number of heterozy- gous loci are overlapping for different phenotypic Prob (2n1 + n2 = r n2 = k) Prob(Y=kjX=r)= classes (see table 1), classification of individuals Prob (2n1 + n2 = r) into different heterozygous classes by phenotypes (5) of individuals can be quite error prone. whichmay be written as It is also interesting to note that while the conditional distribution of the number of heterozy- Prob(Y=kIX=r) gous loci given the phenotypic score is independent of the allele frequency (p), when all loci have I r—k r+k identical allele frequencies, the highest number of k, n3 = n Prob [ni =—-—,n2= heterozygous loci may not always occur for modal phenotypes. For example, if the allele frequencies Prob (2n1 + n2 = r) are skewed (i.e., p
Table 1Conditional distribution of the number of heterozygous loci (Y) given genotypic value (X)
Y x 0 1 2 3 4 5 6 7 8 9 10
n—S 0,10 10 1,9 00 10 2,8 0111 00 0889 3,7 00 0333 0'O 0667 4,6 0048 00 0571 00 0381 5 00 0238 00 0635 00 0127 n 10 0,20 10 1,19 00 10 2, 18 0053 00 0947 3,17 00 0158 00 0842 4, 16 0'009 00 0297 00 0694 5, 15 00 0047 00 0433 0'O 0520 6, 14 0003 00 0130 00 0520 00 0347 7, 13 00 0022 00 0260 00 0520 00 0198 8, 12 0002 00 0080 00 0'400 00 0427 00 0091 9,11 00 0015 0•0 0200 00 0480 00 0274 0.0 0031 10 0001 00 0068 00 0364 00 0437 00 0125 00 0005
In analysis of data of this type, individuals are often classified into modal type and extreme type by grouping them into classes by phenotypes (e.g., Livshits and Kobyliansky, 1984; 1985). For example, if the individuals are classified into three classes: E0 (Xmean—067sd), M(mean— 067sd
20
15 > 0 3 a > 0U C 3. 0 >. 0 C -r0 0. a. 0 C C 3 0 -C 05 a.
0 2 4 6 8 10 0 2 4 6 8 10 Numbero Heterozygous bc Iper Individual
Figure2 Relationship of the number of heterozygous loci with (a) mean and (b) standard deviation of the phenotypic score of a polygenic trait with complete heritability. For these computations the trait is assumed to be controlled by 10 loci at each of which the allele frequencies of two alleles are the same (p).
decreases with k and the mean (,ak)approaches roughly equal to (L/N)'2, although the exact the central value as k increases, the frequency of value is dependent on the mean and variance in central classes of phenotypes increases as the heterozygosity values over N loci (Chakraborty, individuals exhibit more heterozygosity. There- 1981). Since the fraction, L/N, in all empirical fore, we conclude that when heterozygosity of studies is very small, Chakraborty asserted that the individuals reflect heterozygosity at the loci heterozygosity of an individual determined by the governing a quantitative trait, the above two traditional biochemical markers does not provide observations are direct consequences of the addi- an accurate indicator of the individual's genomic tive allelic effects of the loci, and no selective heterozygosity. differential is necessary to explain these findings. Zouros and Foltz (1986) argued that the assumption of random samples of loci being scored by the different biochemical techniques may not HETEROZYGOSITYAT BIOCHEMICAL LOCI AS A be appropriate. In fact, they argue that the REFLECTION OF GENOMIC HETEROZVGOSITY empirical correlation most likely reflects the quantity Inrelation to the studies of heterozygosity and variability of morphological or fitness related traits, commonly only one to about a dozen bio- [ H/Hi], (12) chemical loci are used (see e.g.,Zourosand Foltz, 1987 and the cited references in that review). If where I-I, is the heterozygosity at the ith locus. In these loci are not directly involved in determining other words, the fraction of total heterozygosity at the trait in question, one might ask how well does loci affecting the character accounted by the scored the individuals' heterozygosity detected by these loci can be used as an indicator of how well the loci reflect the genomic heterozygosity of the heterozygosity at the biochemical loci reflects the individual? Mitton and Pierce (1980) and individual's genomic heterozygosity. It is true that Chakraborty (1981) addressed this problem. in heterozygosity vs. morphological diversity Under the assumption that if L loci from a studies only the highly polymorphic loci are collection of N loci are randomly sampled, employed. But Zouros and Foltz's assertion that Chakraborty's analytical treatment shows that the electrophoretic techniques detect the polymorph- correlation between the heterozygosity measured isms with highest heterozygosities does not seem by L loci and the heterozygosity for all N loci is to be correct. Even if the sample of L loci represents HETEROZYGOSITY AND PHENOTYPIC VARIANCE 25 the most heterozygous loci of the genome, it can trophoresis per codon amounts to 00016, which be shown that the correlation between an is close to the lower range of the estimate obtained individual's measured heterozygosity (at L loci) from nucleotide diversity measures. We may there- and the genomic heterozygosity is not given by fore conclude that there is no solid evidence that expression (12), but by the expected value of expression (13) is much larger than (L/N)1'2, as claimed by Zouros and Foltz (1987), and hence IL IN 11/2 prediction of genomic heterozygosity from a survey I H,(l—H1)/ H(l—H1) i Li=i I i=i of small number of highly polymorphic elec- trophoretic loci may not be appropriate. Thereare two types of evidence suggesting that the expression (13) is not anywhere close to one. First, in the statistical analysis of protein poiy- DISCUSSION morphism in natural populations detected by elec- trophoresis, Fuerst etaL(1977) showed that the Thetheory discussed above shows that: (a) the distribution of heterozygosity values across the classification of individuals into different heterozy- studied electrophoretic loci agrees with the expec- gosity classes by phenotypes of a polygenic trait ted distribution under the assumption that these can be quite error prone, and (b) even if the under- are random samples from the genome. Hence, the lying loci of polygenic traits are biochemically average heterozygosity at the population level is detectable, the negative relationship between well estimated when a sufficient number of elec- heterozygosity of individuals and phenotypic vari- trophoretic loci are used. This in turn suggests that ance, and a positive relationship between the the expected values of the individual terms in the frequency of modal class and individuals' numerator and denominator of expression (13) are heterozygosity can be explained by additive allelic roughly equal for the general electrophoretic sur- effects of polygenic traits. Evidence is also pro- veys. If electrophoresis detected most of the vided that suggests that from the electrophoretic genomic polymorphism, this would not have been markers an individual's genomic heterozygosity the case. Hence, the assertion that most of the may not be predicted accurately. Then, the ques- genomic variation is revealed by the surveyed loci tion is how do we explain the observed relationship does not seem to be correct. between phenotypic variance and frequency of Second, even though at present only a small phenotypic classes with electrophoretic heterozy- number of studies have been made regarding the gosity without invoking interaction of loci that genetic diversity at the nucleotide level in natural segregate independently of each other? populations of various organisms, the current esti- To explain these observations by the hypothesis mate of nucleotide diversity is roughly of the order of developmental homeostasis would require pre- of 0002 —0020on a per site basis (Kreitman, 1983; cise estimation of an individual's genomic Chakravarti et aL, 1984; Yager et al., 1984). heterozygosity by the surveyed loci. This is not the However, since substitutions at the third position case, as discussed above. The other two popular of a codon are roughly twice as frequent as those hypotheses have been overdominant selection at at the first or second positions (Nei, 1983), we can the electrophoretic loci or selection at closely assume that the nucleotide diversity for the first or linked protein loci (associative overdominance). second position of the order 00015-0015, and The later hypothesis would reflect that heterozy- that for the third position is 0003 —003.Further- gosity at the protein loci may have a synergistic more, if we assume that the probability of a random effect with that at the loci governing the morpho- nucleotide change causing an electrophoretically logical trait. At this point, it might be mentioned detectable amino acid change is about 028 for the that two sets of independently segregating loci may first position, one-third for the second position, show correlated heterozygosity in individuals of a and only one-twelfth for the third position substructured population. Nei and Li (1973) (Kimura, 1983), the estimated heterozygosity per showed that significant linkage disequilibrium codon detectable by electrophoresis amounts to might be produced at independently segregating 00012 to 0012. On the other hand, in the general loci in a substructured population. Sinnock (1975) electrophoretic surveys the average heterozygosity showed that the two locus Wahiund effect results in natural populations has been observed to be of in a depletion of frequencies of double heterozy- the order of 047 or less (Nei and Graur, 1984). gotes which can be more than the proportional Assuming that an average protein has roughly 300 decrease of heterozygotes at each individual amino acids, the heterozygosity observed by elec- locus. Thus, one may observe a correlation of 26 H. CRAKRABORTY heterozygosity levels at two unlinked loci for 1985), it would be more troublesome still to invoke individuals in a subdivided population, and hence, the overdominant hypothesis as a general rule. the heterozygosity levels detected by protein loci Furthermore, in examining the allele frequency may be correlated with heterozygosity of loci distributions of some 138 populations of various governing the morphologic trait to some extent, organisms, Chakraborty eta!.,(1980) showed that even if they are independently segregating. Of the allele frequency spectrum is generally U- course, since the observed levels of linkage dis- shaped (which is in accordance with the neutral equilibrium at the surveyed loci in natural popula- model), instead of being bell-shaped orW-shaped, tions is quite low (e.g.,Mukaieta!.,1971, 1974; which is the prediction of the overdominant model Sinnock and Sing, 1972; Langley eta!., 1974),it is (Li, 1978). The observations on activity levels in unlikely that the correlation between heterozygos- heterozygotes and homozygotes of some enzymes ity levels at the biochemical loci and the loci are also against the overdominant hypothesis (see governing morphologic variation can be explained e.g.,Harris,1975; Kacser and Burns, 1981). by multi-locus Wahiund effects alone. The associative overdominance hypothesis, as Evidence is now being accumulated suggesting metioned earlier, may be an easier explanation for that electrophoretically determined biochemical the correlation between the number of heterozy- variation has measurable physiological con- gous loci and fitness related traits. This hypothesis sequences, some of which may modulate the apparently has its origin due to Jones (1917), as phenotypic traits. Koehn eta!.(1983) reviewed documented by Nei (1987). It is well known that extensive experimental evidence of this suggestion. in a finite population significant linkage disequi- In humans, Orr eta!.(1981)showed that variation librium between some traits may be produced by in traits like serum cholesterol is affected by genetic drift (Hill and Robertson, 1968; Sved, genotypes at the ABO, Haptoglobin, Gamma 1968) and this would cause associative overdomin- globulin, and Secretor loci. Boerwinkle eta!.(1986) ance. Ohta's (1971) study also shows that when a showed that several biochemical traits indeed protein locus with two alleles is linked with explain substantial proportions of genetic variabil- deleterious genes, heterozygotes at the protein ity of many quantitative traits. At the DNA level locus may demonstrate higher fitness than homozy- too, restriction site polymorphisms are shown to gotes. Nevertheless, unless the loci governing the explain significant component of variations of phenotypic traits are determined, these explana- some physiological traits (e.g.,Haniseta!., 1985). tions cannot be tested directly. Thus, it is not totally unreasonable to assume that Finally, it should be stated that not all reports major genes residing at close linkage distance from of correlation between heterozygosity and fitness some of the electrophoretically determined related traits are real. For example, Kobyliansky markers are the factors that at least partially control and Livshits (1985) claimed evidence of heterozy- morphological variability. If this is the case, then gous advantages in human populations in terms the observation of an association of heterozygosity of fertility in connection with five morphological with phenotypic variability may truly be the reflec- characters: weight (W), stature (S), hand width tion of the additive effects of the loci underlying (HAW), bigonial diameter (BIG), and ear width these traits. (EW). In showing these, they have classified 230 Several authors favour the hypothesis of over- spouses from Mexican families into three classes dominant selection to explain the association of (low= 1, intermediate=2, and high=3), depend- phenotypic variance and biochemical heterozygos- ing on their phenotypic scores. ity (e.g.,Mittonand Grant, 1984 and the cited Assuming that these traits are polygenic and references of their review). This hypothesis poses heritable, they divided the parental data into nine a number of problems regarding other facets of mating types. On the supposition that the expected protein polymorphism data. For example, if over- heterozygosity in the offspring of these mating can dominant selection operates on many protein loci, be arranged in ascending order, they reported the the average heterozygosity is expected to be much mean and s.d. of the number of living children higher than the one predicted by the neutral muta- from each mating type. Their analyses indicate that tion theory (Maruyama and Nei, 1981; Nei and fertility, as determined by number of living child- Graur, 1984). But the observed levels of heterozy- ren, is positively correlated with the heterozygosity gosities in various organisms are generally much when parental phenotypes as well as expected smaller than their neutral expectations. If the heterozygosity of the offspring are taken into neutral mutation model is in trouble to explain account. From their data (summarised in table I this (as suggested by Livshits and Kobyliansky, of their paper), a simple analysis of variance may HETEROZYGOSITY AND PHENOTYPIC VARIANCE 27 be conducted to see if the nine mating types show the additive allelic effects at loci that are directly any significant differences in the fertility levels, the involved, or at disequilibrium with the trait may result of which is negative for each trait they cause such association may be sufficient to explain examined, contrary to their findings. Furthermore, some of these observations. In this regard the if we group the mating types by the expected measured genotype approach (Boerwinkle et al., heterozygosity of the offspring (which is error 1986) may help substantially to resolve the con- prone, as shown in this paper), the three groups troversy regarding the underlying mechanism that of matings that give rise to expected heterozygosity might cause an association between phenotypic of OO (from matings lxi, and 3x3), O5 (from variability and biochemical heterozygosity. matings 1x2,2xl,2x3,3x2,and2x2),and1O (from matings 1 x 3 and 3 x 1) are also statistically homogeneous with regard to mean fertility. Acknowledgements This work is supported by U.S. Public Table 3 shows these computations from the Health Service research grants from the National Institutes of Health and the National Science Foundation. Comments from data presented in Kobyliansky and Livshits (1985). Drs W. J. Schull, M. Nei, C. Stephens, and C. L. Hanis are These results are in direct contradiction with the greatly appreciated. assertion of their paper, but are in accordance with the findings of Chakraborty et a!. (1986) that fer- tility is not affected by heterozygosity of the individuals determined from other loci, even REFERENCES though in the study of Chakraborty et a! (1986) BOERWINKLE, E., CHAKRABORTY,R. AND SING, C. F. 1986. no effort is made to determine the expected The use of measured genotype information in the analysis heterozygosity of the offspring of mothers of of quantitative phenotypes in man. Ann. Hum. Genet., 50, different genotypes. 181—194. In conclusion, we may state that even when the CHAKRABORTY, R. 1981. The distribution of the number of observations on associations between biochemical heterozygous loci in an individual in natural populations. Genetics, 98, 461-466. heterozygosity and morphologic traits are real, CEIAKRABORTY, R., FERRELL, R. E., BARTON, S. A. 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Table 3Relationship between parental fitness and expected heterozygosity among children in a study from a Mexican population*
Mating types of parentst
(a) (b) (c) F-Ratio Traitt n mean s.d. n mean s.d. n mean s.d. df F
W 34 5'20 221 165 586 241 29 621 213 (2, 225) 1'58 5 33 5'43 188 179 587 245 17 583 205 (2,226) 0'50 HAW 45 525 194 144 5'93 244 31 5'81 257 (2,217) 1'44 BIG 31 565 2'25 164 5'66 225 26 669 247 (2, 218) 234 EW 23 5.30 1'99 178 5'77 239 19 6'53 2'18 (2, 217) 1'45 EH 24 5'59 2'45 179 5'79 233 19 6'll 2'51 (2,219) 0'26 MN 28 582 218 145 577 238 24 584 239 (2, 194) O'Ol CI 27 618 284 171 585 233 29 533 206 (2, 224) 0'94 * Datataken from table 1 of Kobyliansky and Livshits (1985). The parental fitness is measured by the number of living children from each family. tme mating types are classified by the phenotypic scores of each spouse: (1) with phenotypic score in the lowest 25 percentile; (2) with phenotypic score in the middle 25 to 75 percentile; and (3) with phenotypic score in the upper 25 percentile of the distributions. The three mating types are: (a) I x 1 and 3 x 3 (the matings for which the expected heterozygosity of the offspring is 0); (b) 1 x 2, 2 x 1, 2 x 3, 3 x 2, and 2 x 2 (the matings for which the expected heterozygosityin the offspring is 0'S); and (c) 1 x 3 and 3x I (the matings for which the expected heterozygosity in the offspring is 1). The traits are: W (weight); S (stature); HAW (Hand width); BIG (Bigonial diameter); EW (ear width); EH (ear height); MN (menton-nasion index); and CI (cephalic index). 28 R. CHAKRABORTY
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